Wave Motion

What is a wave? If you've ever seen the ripples caused by dropping a pebble in a puddle of water, then you have seen a wave. A wave is a disturbance that transfers energy. It was thought for a long time that a medium was required for the transfer of energy to take place, but today we know that isn't true. Waves come in different forms: water waves, sound, light, even earthquakes propagate in waves. We will make two distinctions:

  • Mechanical waves are waves that require a medium to transfer. Sound is a good example.
  • Electromagnetic waves are waves that can transfer through a vacuum. Light, radio, and x-rays are examples of electromagnetic waves.

For the time being, we will concern ourselves with mechanical waves. Within that context, we will also define transverse waves and longitudinal waves. A transverse wave is wave with an up and down motion. To be more specific, as the energy passes along the wave, particles in the medium travel in a direction perpendicular to the direction of energy motion. Our pulse on a string is a good example of this. Actually, so is the fan "wave" at a baseball game. A longitudinal wave has the particles in the medium bouncing back and forth in the same direction as the energy is being transferred, similar to dominos. Sound travels this way, with the molecules being compressed and expanded transferring the energy. For more on longitudinal and transverse waves, click on http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html.

Let's start with a simple wave - a single pulse. Something like this occurs when you take a rope or a slinky and give it a quick shake. The pulse will travel along the length of the rope. When it reaches the end, it will actually be reflected back. A single pulse like this is called a pulse wave.

When a wave is caused by a series of disturbances or pulses, we say it is continuous or periodic. The energy will come from a series of vibrations or oscillations. The vibration could be produced by a number of things that you may be familiar with: shaking your hand on the string, dropping the pebble in the water, plucking a guitar string, or turning on a motor. Anything that produces a regular pattern of vibrations will produce a regular series of waves.

When we talk about waves, we use some pretty specific terms to define what we mean. A crest is the high point on a wave - it is the point of maximum disturbance, or an area of maximum density and maximum pressure. Similarly, the trough is the low point on a wave. The trough is a low density, low pressure region on a wave. (Longitudinal waves are sometimes termed pressure waves or density waves.) The distance between equilibrium and the crest, or equilibrium and the trough, is called the Amplitude. Amplitude is the amount of energy in the wave, and we always talk about it in terms of an absolute value. The greater the amplitude, the more energy a wave has. In a sound wave, for example, amplitude relates to the volume, or how loud the sound is.

The distance between two crests, or between two troughs, or between any two alike points (one complete cycle) on a wave is called the wavelength and is given by the Greek letter lambda (l). Wavelength is measured in meters, although depending upon our type of wave, we will probably use prefixes, such as micrometers and nanometers. Light is generally denoted in nanometers (nm).  Two quantities are used to talk about the timing of a wave: The period (T) and the frequency (f). Period is the amount of time required for a wave to make one complete cycle. It is measured in seconds. Frequency is the number of cycles in one second, or the number of times that a wave repeats itself in one second. The units of frequency are cycles per second, which we also call Hertz (Hz). For example, the frequency of an FM radio station is in the range of 100 Megahertz (Mhz). That is to say, an FM wave oscillates 100 million cycles every second. AC electricity that we have in our homes oscillates at 60 Hz.

The fundamental frequency that an object resonates at is called the first harmonic. A harmonic is an integer multiple of the fundamental frequency and is used to show the number of standing waves that can appear in a fixed interval. We'll talk more about these when we talk about sources of sound. An overtone is basically another
name for a harmonic, except the numbering is a little different. The 2nd harmonic is the 1st overtone. So, for example, if we have a resonance at a fundamental frequency of 400 Hz, then the 2nd harmonic is at 2(400 Hz) = 800 Hz. This would also be the first overtone.

The model for a wave is the sine function, and the equation of a wave can generally be given in the form 


where A is the amplitude, t is the elapsed time, q is the phase shift, and w defines the period (period = 2p/w).

The speed of a wave (v) is related to frequency and wavelength, and is given by the equation



Speed of a wave is an interesting topic all to itself. The speed of a wave depends upon the medium it is in. For a given medium, a wave's speed does not change. So if frequency increases, then wavelength must decrease. But, when a wave travels from one medium to another, such as when a beam of light enters the water from air, the frequency of the wave remains constant, but the speed of the wave, and the wavelength both change. We'll
discuss this more when we talk about refraction of waves.

There are several formulas that we will use to determine the speed of a wave in a particular medium. The three most common ones for our course are:

Used to find the speed of a wave in a wire, such as a guitar or piano string. FT is the tension in the wire, and m/L is the mass per unit length (or linear density) of the wire.

Used to find the speed of a longitudinal wave in a long, solid rod. Eis the elastic modulus of the material and r is the density of the material. We will get these from tables.

Used to find the speed of a longitudinal wave in a gas or liquid. Bis the bulk modulus of the material and r is the  density of the material. We will get also these from tables.